The Fibonacci Sequence and the Golden Ratio

1, 1, 2, 3, 5, 8, 13, 21, 34, …

These numbers form one of the most recognizable sequences in the world. It is known as the Fibonacci sequence and it’s named after Leonardo Pisano, who was also known as the 13th century mathematician Fibonacci. The story goes, Fibonacci was working out the growth rate of rabbits in ideal conditions. Moreover, he attempted to state exactly how many pairs rabbits exist after one year with the following assumptions.

  • We introduce 1 pair of baby rabbits at month 1.
  • The rabbits reach maturity in 1 month and are guaranteed to mate and produce 1 pair of babies.
  • No rabbits will perish

So then we get at month 1 we introduce a pair of baby rabbits. In 1 month they reach maturity and still haven’t produced hence we have 1 pair. The following month our rabbits produce a pair and so we have 2 pairs. In the third month, the original pair produces a new pair and the second pair reaches maturity. That is, we have 3 pairs. We continue in this way and obtain the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 pairs of rabbits in the 12th month. One may have seen by now that the nth term in the Fibonacci sequence, F_n is obtained by adding F_{n-2} + F_{n-1}. This sequence is very popular and can indeed be found in nature. First, let us look at the Fibonacci spiral.

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Each of these squares has the designated number as its side length. The spiral is then drawn in the obvious way. Now we shall see where this sequence, and spiral occur in nature.

This sequence is also common in modern architecture and art. As the title of this post suggests, the Fibonacci sequence also has a close relationship with the Golden ratio. Take a look back up at the spiral, particularly the rectangle. If at each step we take the ratio of the side lengths (largest over smallest) and we continue to build this rectangle all the while taking these ratios we obtain the sequence

1, 2, 3/2, 5/3, 8/5, 13/8, … Which can be written as a sequence of approximates,

1, 2, 1.5, 1.667, 1.6, 1.625, …

We will prove that this ratio sequence indeed converges to the Golden Ratio \Phi = \frac{1 + \sqrt{5}}{2} \approx 1.61803.... We first see that \Phi = 1 + \frac{1}{\Phi}.

Now is we call the nth term in this ratio sequence to be R_n we can deduce R_n = 1 + \frac{1}{R_{n-1}}. Now we compute

|R_n - \Phi| = \left| \left( 1 + \frac{1}{R_{n-1}} \right) - \left( 1 + \frac{1}{\Phi} \right) \right| = \left| \frac{1}{R_{n-1}} - \frac{1}{\Phi} \right| = \left| \frac{\Phi - R_{n-1}}{R_{n-1} \Phi} \right| \leq \frac{1}{\Phi} \cdot |R_{n-1} - \Phi|

Note that we can repeat this process inductively until we find,

|R_n - \Phi| \leq \left( \frac{1}{\Phi} \right) ^{n-1} |R_1 - \Phi|.

To conclude the proof we notice that \frac{1}{\Phi} < 1 implies \lim_{n \to \infty} \left( \frac{1}{\Phi} \right)^n = 0. This tells us that the ratio sequence indeed converges to the Golden Ratio. BUT! Notice that the only property we used of the sequence was that R_n = 1 + \frac{1}{R_{n-1}}. So pick ANY two real numbers and set them to be S_1 and S_2. Form the sequence by the rule S_{n-2} + S_{n-1}. Then form the ratio sequence T_n = \frac{S_{n+1}}{S_n}. Then we see that

1 + \frac{1}{T_{n-1}} = 1 + \frac{1}{\frac{S_n}{S_{n-1}}} = 1 + \frac{S_{n-1}}{S_n} = \frac{S_n + S_{n-1}}{S_n} = \frac{S_{n+1}}{S_n} =  T_n.

Therefore we conclude that the limit of any sequence formed by this rule has a limit that is indeed the Golden Ratio. I encourage the interested reader to explore the inverse sequence \frac{R_{n-1}}{R_n}

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